APS/PRE Stability of a planetary climate system with the biosphere competing for resources Sergey A. Vakulenko, 1,2 Ivan Sudakov, 3,* Sergei V. Petrovskii, 4 and Dmitry Lukichev 2 1 Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, St.Petersburg 199178, Russia 2 Department of Electrical Engineering and Precision Electro-Mechanical Systems, ITMO University, St. Petersburg 197101, Russia 3 Department of Physics, University of Dayton, 300 College Park, SC 111, Dayton, Ohio 45469-2314, USA 4 School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK * Abstract With the growing number of discovered exoplanets, the Gaia concept finds its second wind. The Gaia concept defines that the biosphere of an inhabited planet regulates a planetary climate through feedback loops such that the planet remains habitable. Crunching ’Gaia’ puzzle has been a focus of intense empirical research. Much less attention has been paid to the mathematical realization of this concept. In this paper, we consider the stability of a planetary climate system with the dynamic biosphere by linking a conceptual climate model to a generic population dynamics model with random parameters. We first show that the dynamics of the corresponding coupled system possesses multiple timescales and hence falls into the class of slow-fast dynamics. We then investigate the properties of a general dynamical system to which our model belongs and prove that the feedbacks from the biosphere dynamics cannot break the system’s stability as long as the biodiversity is sufficiently high. That may explain why the climate is apparently stable over long time intervals. Interestingly, our coupled climate-biosphere system can lose its stability if biodiversity decreases; in this case, the evolution of the biosphere under the effect of random factors can lead to a global climate change. 1 arXiv:1902.07936v3 [nlin.CD] 9 Jun 2020
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APS/PRE
Stability of a planetary climate system with the biosphere
competing for resources
Sergey A. Vakulenko,1,2 Ivan Sudakov,3,∗ Sergei V. Petrovskii,4 and Dmitry Lukichev2
1Institute of Problems in Mechanical Engineering,
Russian Academy of Sciences, St. Petersburg 199178, Russia
2Department of Electrical Engineering and Precision Electro-Mechanical Systems,
ITMO University, St. Petersburg 197101, Russia
3Department of Physics, University of Dayton,
300 College Park, SC 111, Dayton, Ohio 45469-2314, USA
4School of Mathematics and Actuarial Science,
University of Leicester, Leicester LE1 7RH, UK∗
Abstract
With the growing number of discovered exoplanets, the Gaia concept finds its second wind.
The Gaia concept defines that the biosphere of an inhabited planet regulates a planetary climate
through feedback loops such that the planet remains habitable. Crunching ’Gaia’ puzzle has been
a focus of intense empirical research. Much less attention has been paid to the mathematical
realization of this concept. In this paper, we consider the stability of a planetary climate system
with the dynamic biosphere by linking a conceptual climate model to a generic population dynamics
model with random parameters. We first show that the dynamics of the corresponding coupled
system possesses multiple timescales and hence falls into the class of slow-fast dynamics. We
then investigate the properties of a general dynamical system to which our model belongs and
prove that the feedbacks from the biosphere dynamics cannot break the system’s stability as long
as the biodiversity is sufficiently high. That may explain why the climate is apparently stable
over long time intervals. Interestingly, our coupled climate-biosphere system can lose its stability
if biodiversity decreases; in this case, the evolution of the biosphere under the effect of random
factors can lead to a global climate change.
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I. INTRODUCTION
Understanding of the mechanisms and scenarios of climate change as well its current
and potential effects on ecosystems and biodiversity have been a focus of keen attention
and intense research over the last few decades [1–3]. There is a general consensus that
climate change will likely have an adverse impact on the ecological systems and population
communities resulting in species extinction and a considerable biodiversity loss worldwide.
Whilst the top-down effect of climate on ecosystems is thus well established, relatively
little attention has been paid to a possibility of an opposite, bottom-up effect that ecosystems
may have on the climate. The mainstream of research often tends to consider the ecosystems
and population communities as biological actors on the physical stage [4] often disregarding
possible feedback. Meanwhile, in planetary science, there is the concept of Gaia [5] that
postulates the biosphere regulates its planetary climate to mitigate it for its own survival.
While this hypothesis has been introduced quite long ago, current research in planetary
and earth sciences inspires new applications of this hypothesis. The work [6] is shown that
even if a model exoplanet has significant climate perturbations then the Gaia concept is still
acceptable (the original Gaia concept is based on a static planetary climate). Another work
[7] supports the Gaia concept considering Earth’s biosphere stability over climate change
through the existence of climate feedback loops and climate tipping points [8, 9].
In this paper, we present a new mathematical realization of the Gaia hypothesis by a
model of coupled climate-biosphere dynamics. We consider the effect that the biosphere of
a planet may have on a planetary climate by changing the global energy balance through
modifying the planetary albedo.
Modeling of physical processes in the climate system leads to difficult problems, involving
complicated systems of partial differential equations for biological and chemical processes
[2]. There exist climate models with different levels of realism; they can include thousands
and even millions of equations, thousands of parameters to adjust. Usually, one investigates
these models by computer simulations [10]. However, it is difficult to estimate the reliability
of these computations, since it is connected with a difficult mathematical problem on the
structural stability of attractors [11, 12]. The theory of linear response of climate systems
to perturbations [13] is based on the Ruelle theory of linear response for dynamical systems
that holds on the formal hypothesis that the dynamical system is of the type axiom A one.
2
The last fact implies structural stability. However, S. Smale’s A-axiom systems [12] seldom
appear in practical applications. The class of structurally stable systems is very narrow; this
mainly includes systems with hyperbolic or almost hyperbolic behavior. One can expect,
therefore, that the attractors of climate systems are not structurally stable: their topological
structure can change under small perturbations. Therefore, one can expect that they can
exhibit complicated bifurcations under small parameter perturbations. Possibly, an adequate
approach is to take into account random fluctuations and study random dynamical systems.
Indeed, the climate system as a complex system has a large number of interconnected and
interacting subsystems, including the following: the atmosphere, the oceans, the biosphere,
etc. Determination of how the dynamics of these subsystems change to reach equilibria of
the entire system is the main problem of so-called conceptual climate models.
There are different types of conceptual climate models. Many of them are energy balance
climate models and are defined by an ordinary differential equation describing energy con-
servation in the climate system. The most popular model is a zero-dimensional model [14]
based on the theory of blackbody radiation determining global temperature changes due to
the difference in incoming and outgoing solar radiation. This difference may be caused by
changing of control parameters: surface albedo, greenhouse gas emission, and even the solar
constant. The equilibria and the ideas how to find them by the bifurcation theory tools can
be found here [15].
In the context of Gaia hypothesis [8, 9] here arises a key question: Why does climate
stays stable over long time intervals (e.g. hundreds of thousands of years)? To answer this
question, we consider conceptual climate models where the dynamical variables may be
decomposed as slow and fast modes. Then for large times fast mode dynamics is captured
by the slow dynamics on a stable slow manifold of a slow-fast system. The slow variables
determine a long-term climate evolution under external factors whereas the fast modes may
be associated with rapid factors.
The paper is organized as follows. In the next section, we introduce a planetary cli-
mate model with a biosphere component that arises from coupling between the conceptual
zero-dimensional global energy balance model of climate dynamics and a generic ecosys-
tem dynamics model (a multispecific population system living on multiple food sources).
In Section III, we consider a general class of systems to which our model belongs and dis-
cuss the stability of those systems. We then show in Section IV that, in the case of our
3
climate-biosphere model, a planetary climate remains stable with regard to a variation of
the ecosystem model parameters as long as biodiversity is sufficiently large but it can lose
stability (hence potentially resulting in regime shifts and a global climate change) if the
number of species is small. A discussion and conclusions can be found in the last section.
II. THE MODEL
The energy balance system is one of simplest climate models. It is defined by the following
equation [15] :dT
dt= λ−1
(−eσT 4 +
µ0I0
4(1− A)
), (1)
where λ is thermal inertia, T is the averaged surface temperature, t is time, and A is
the albedo of the surface. The left term characterizes the time-dependent behavior of the
climate system. On the right hand side, the first term is the outgoing emission and the
second term represents the incoming star’s radiation. Generally, incoming radiation to the
planetary surface from a star is modified by a parameter, µ0, to allow for variations in the
stellar irradiance per unit area, I0 (the solar constant in case of the Earth), or for long-term
variations of the planetary orbit [16]. On the other side, the outgoing emission depends on
the fourth power of temperature, the effective emissivity e and a Stefan-Boltzmann constant
σ.
This model can be coupled with the modeled biosphere’s dynamics as follows. The
complete averaged albedo A can depend on the biosphere state. For simplicity, we mostly
focus our analysis on a single global ecosystem competing for several resources. We consider
the following classical model:
dxidt
= xi(−µi + φi(v)− γi xi), i = 1, . . . ,m, (2)
dvkdt
= Dk(Sk − vk)−M∑i=1
bki xi φi(v), k = 1, . . . , n, (3)
cf. [17, 18], where x = (x1, x2, ..., xn) are the species abundances, m 1, and v = (v1, ..., vn)
the resource concentrations. Here µi are the species mortalities, Dk > 0 are resource turnover
rates, and Sk is the supply of the resource vk, φi is the specific growth rate of species as a
function of the availability of the resource (also known as MichaelisMenten’s function). The
coefficients γi > 0 define self-limitation effects [19]. We assume that each of the resources
4
vk, k = 1, . . . , n, is consumed by all species so that the content of k-th resource in the i-th
species is positive bik > 0.
We consider general φj which are bounded, non-negative and Lipshitz continuous
0 ≤ φj(v) ≤ C+, |φj(v)− φj(v)| ≤ Lj|v − v|, (4)
i.e., φk have a minimal smoothness, they are bounded and non-negative. The last restriction
means that species consume resources.
Moreover, we suppose
φk(v) = 0, for all k, v ∈ ∂Rm+ (5)
where ∂Rm+ denotes the boundary of the hyperoctant Rm
+ = v : vj ≥ 0, ∀j. Moreover, we
suppose that∂φk(v)
∂vj≥ 0, for all k, j, v ∈ ∂RM
+ . (6)
This assumption means that as the amount of the j-th resource increases all the functions
φl also increase.
Conditions (4) and (5) can be interpreted as a generalization of the well known von Liebig
law, where
φk(v) = rk min v1
Kk1 + v1
, ...,vm
Kkm + vm
(7)
(cf. [17]) where rk and Kkj are positive coefficients, and k = 1, ...,M . The coefficient rk is
the maximal level of the resource consumption rate by the k-th species and coefficients Kki,
i = 1, ...,M define the sharpness of the consumption curve φk(v).
A simple way to couple climate subsystem (1) and the modeled biosphere defined by
(2) and (3) is to suppose that the resource supply parameters Sk depends on the surface
temperature T . Moreover, we can suppose the albedo is a linear function of xi:
A = A(x) = A0 +m−1
m∑j=1
cjxj. (8)
Finally, we obtain the following climate-biosphere system
dxidt
= xi(−µi + φi(v)− γi xi), i = 1, . . . ,m, (9)
dvkdt
= Dk(Sk(T )− vk)−m∑i=1
bki xi φi(v), k = 1, . . . , n. (10)
5
dT
dt= λ−1
(−eσT 4 +
µ0I0
4
(1− A0 +m−1
m∑j=1
cjxj
)). (11)
As an example, let us consider a model planet where the surface significantly covered by
ice [20] and the ice-albedo feedback is the main regulator of the planetary climate dynamics
[21]. Let the area of some region of the planet be Sarc, the area occupied by ice be Sice
while the free ice area be Sfree [22], where Sfree = Sarc − Sice. One can suppose that
different species coexist in free ice domain and the averaged albedo of this domain is a linear
combination of contributions of different species. Then we obtain